Download Prices and Exchange Rates In frictionless markets

Prices and Exchange Rates
In frictionless markets, freely tradable goods should have the
same price anywhere:
P$ × S = PU
P $ ≡ price in US$
S
≡ Exchange rate in yen per dollar
P U ≡ Price in Japanese yen
Purchasing power parity (PPP) exchange rate:
PU
S =
P$
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The hamburger strandard (April 17, 2001):
Price of a Big Mac: C$ 3.33 and US$2.54.
PPP exchange rate (C$/US$):
3.33
= 1.31C$/US$
S =
2.54
Actual exchange rate: 1.56 C$/US$.
According to the PPP theory, Canadian dollar is undervalued by
1.31
− 1 = − 16%
1.56
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In a less extreme form, the PPP theory should hold for baskets
of goods.
P I U ≡ price of a basket of goods in yen
P I $ ≡ price in US$ of the same basket
PPP exchange rate between Japan and US:
P IU
S =
P I$
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Relative Purchasing Power Parity
Relative changes in prices between two countries over a period
of time is what determines the changes in exchange rates over
the same period.
If the spot exchange rate between two countries starts in equilibrium, any change in the differential rate of inflation between
them tends to be offset over the long run by an equal but opposite change in the spot exchange rate.
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P It$ ≡ US price index at time t
P ItU ≡ Japanese price index at time t
π $ ≡ US inflation from t to t + 1
π U ≡ Japanese inflation from t to t + 1
St ≡ exchange rate at time t (U/$)
Assuming purchasing power parity,
St =
P ItU
P It$
and
P ItU (1 + π U )
1 + πU
St+1 =
= St ×
$
1 + π$
P It (1 + π $)
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Percent change in S from t to t + 1:
U
St+1
−1 =
St
St × 1+π $
1+π
St
−1
1 + πU
=
−1
$
1+π
πU − π$
=
1 + π$
∝ πU − π$
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Example
Honda’s price per vehicle: U4,000,000
S = U125/$
Expected inflation in Japan: 1%
Expected inflation in U.S.: 3%
Assuming PPP, what is the expected spot exchange rate at the
end of the year?
1 + πU
Send = S ×
1 + π$
1.01
= U125/$ ×
1.03
= U122.57/$
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Real and Nominal Exchange Rates
Suppose PPP holds:
$1.50/£ =
$1, 500/U.S. good
£1, 000/British good
and thus
$1.50/£
= 1 British good per U.S. good,
$1, 500/£1, 000
i.e. identical goods trade on a one-for-one basis.
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Real Exchange Rate
The units of the real exchange rate between U.S. and U.K. is
U.S. goods
British goods
or
British goods
U.S. goods
When exchange rates change by the same percentage as relative
prices, real exchange is constant.
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Real Exchange Rate
Real Exchange Rate =
Nominal Exchange Rate
PPP Exchange Rate
SR ($/£) =
SN ($/£)
P I $/P I £
If SR ($/£) > 1, US$ is undervalued vis-à-vis the £.
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Real Exchange Rate Index
Real Exchange Rate Index =
Nominal Exchange Rate Index
PPP Exchange Rate Index
ER (FC/$) =
EN (FC/$)
C FC/C $
If changes in exchange rates offset differential inflation,
ER = 100.
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PPP and Transaction Costs
Transportation Costs
Helliwell (1998) Canadian provinces are 12 times more likely to
trade merchandises and 40 times more likely to trade services
among themselves then with closer American states
The border matters
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Tests of Purchasing Power Parity
PPP holds well over the very long run but poorly for shorter time
periods
The theory holds better for countries with relatively high rates
of inflation and underdeveloped capital markets
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Example
Businesses and consumers tend to abandon their currency in a
hyperinflationary economy.
Suppose a TV set is priced 1,200 pesos in Mexico, reflecting a
$400 TV price and 3 peso/$.
If the peso devalues to 6 peso/$, the price of a TV set becomes
2,400 pesos
PPP holds immediately since merchants use PPP to calculate
their prices
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Exchange Rate Pass-Through
Incomplete exchange rate pass-through could explain why an
exchange rate can deviate from its PPP-equilibrium level.
Example: BMW produces automobiles in Germany and pays all
production expenses in euros. When esporting to the US, the
price should be
P$
= P€
×S
BMW
BMW
$
should
If the euro appreciates by 10% over the US$, then PBMW
also increase by 10%.
$
If instead PBMW
increases by less than 10%, the pass-through is
partial.
BMW has an incentive to limit the pass-through.
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Example: Suppose exchange rate is $1/€.
$
€
PBMW
= PBMW
€
If the euro appreciates 20% versus the dollar and PBMW
=
€35, 000,
$
PBMW
= 1.2 × 35, 000 = $42, 000.
$
If, instead, PBMW
= $40, 000, the pass-through is
$
€
PBMW
/P
.14
40, 000/35, 000
,2 BMW,1
=
=
= 70%
(S2 − S1)/S1
.20
.20
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Pass-through may depend on the price elasticity of demand
p =
∆Qp
∆p
Goods with inelastic demand, i.e. |p| < 1, should be associated
with a higher degree of pass-through
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Example
Honda’s price per vehicle is U4,000,000, S = U125/$
Expected inflation in Japan: 1%, Expected inflation in U.S.: 3%
(a) Assuming PPP and 100% pass through, what is the expected
price of a Honda in US$ at the end of the year?
$
U
Pend
× Send = Pend
1+π U
U
U
=
1
+
π
P
×
S
×
$
beg
end
1+π
1+π $ P U
beg
$
Pend
=
S
P$
$
Pend
= 1.03×4,000,000
= $32, 960
125
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(b) Assuming PPP and 60% pass through, what is the expected
price of a Honda in US$ at the end of the year?
Sbeg = U125/$ ,
Send = U122.57/$
Japanese yen appreciates
1
1
−
Send
Sbeg
1
Sbeg
=
Sbeg − Send
Send
125 − 122.57
= 1.944%
=
125
$
versus the dollar. With a 60% pass through, Pend
will increase by
.6 × 1.944% = 1.1664%
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Interest Rates and Exchange Rates
The Fisher Effect
1 + r =
1+i
1+π
⇒
i = r + π + rπ
r ≡ real exchange rate
i ≡ nominal exchange rate
π ≡ inflation rate
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The International Fisher Effect
i$ ≡ nominal interest in the US
iU ≡ nominal interest rate in Japan
S1 ≡ exchange rate (in U/$) at the beginning of the period
S2 ≡ exchange rate at the end of the period
Return should be the same in both markets
S1
(1 + iU ) = 1 + i$
S2
⇒
S1
1 + i$
=
S2
1 + iU
Subtracting one on both sides:
S1 − S2
i$ − iU
=
S2
1 + iU
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Empirical tests provide some support to this thesis.
This model does not take into account foreign exchange risk.
Speculation may also creates distorsions in currency markets.
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The Forward Rate
The forward rate is an exchange rate quoted for settlement at
some future date: 30, 60, 90, 180, 270, 360 days
Invest $1 in the U.S. a the rate i$ and enter a forward contract
that exchanges US$ for Swiss Francs (SF) at the rate F SF/$.
Exchange US$ for SF today at the spot rate S SF/$ and invest
it is Switzerland at the rate iSF.
Should end up with the same number of SF:
$
1 + i F SF/$ =
1 + iSF S SF/$
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SF/$
F90
≡ 90-day forward rate for the SF/$ exchange rate
ic ≡ annual interest rate in (currency c)-denominated deposits
SF/$
F90
SF × 90
1
+
i
360
= S SF/$ ×
1 + i$ × 90
360
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Interest rate parity
Spot and forward rates are considered to be at interest rate parity
if
SF/$
n
1 + iSF × 360
Fn
=
n
SF
/$
S
1 + i$ × 360
Forward premium:
SF/$
fn
SF/$
S SF/$ − Fn
=
SF/$
Fn
=
n
i$ − iSF × 360
1 + iSF × n
360
Currency with a lower interest rate sells forward at a premium
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Covered Interest Arbitrage (CIA)
If spot and forward rates are not at interest rate parity, this
creates covered interest arbitrage opportunities.
These opportunities will continue until spot and forward rates
reach interest rate parity.
Political risk: Application of capital controls
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Uncovered Interest Arbitrage (UIA)
Money is borrowed from countries with low interest rates and
invested in countries with high interest rates, without covering
with a forward contract.
Expected profits calculated using expected spot rates: foreign
exchange risk.
Example: The “yen carry trade”
Borrow yen at extremely low interest rate (0.4%)
Change for dollars, invest the dollars at 5%
Change dollars for yen to repay loan.
If the Yen strenghtens by more than the interest rate differential,
investors lose money
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Forward Rate as an Unbiased Predictor of the Future Spot
Rate
Hypothesis:
Et[St+k ] = Ft,t+k
Future spot rates should be distributed around the forward rate
if hypothesis is true.
If markets are not efficient, then the forward rate is not an unbiased predictor of the future spot rate.
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Asset Market Approach to Forecasting
Exchange rates depend not only on differences in interest rates
but also on a country’s outlook for economic growth and profitability.
Exchange rates depends on the willingness to hold some currencies.
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Problem 1. Polish Zloty
On year ago the exchange rate between Polish zloty and the
U.S. dollar was Z3.8000/$. Since then the zloty has fallen 14%
against the dollar. Price level in the United States have not
changed, but Polish prices have gone up 7%.
a. What is the nominal exchange rate today?
b. What should be the exchange rate today, based on purchasing power parity and assuming last year was normal?
c. Did the Polish zloty depreciate or appreciate in real terms
relative to the dollar? By what percentage?
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Problem 2. Pinot Noir
Pinot Noir wine is produced in the states of California (U.S.A.)
and New South Wales (Australia). Equivalent bottles of Pinot
Noir sell in the United States for US$22 and in Australia for
A$34.
a. According to the PPP theory, what whould be the US$/A$
spot rate of exchange?
b. Suppose the price of Pinot Noir is expected to rise to $27
over the next year while the price of a comparable bottle of
Australian wine is expected to rise to A$44. What should be
the one-year forward US$/A$ exchange rate?
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c. Given your answers to (a) and (b) above, and given that the
current interest rate in the U.S. is 5% for notes of a one-year
maturity, what would you expect current Australian interest
rates to be?
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Problem 4. Covered Interest Arbitrage (U/$)
Ayako Miyake is a currency arbitrager for Atsugi Bank in Kanagawa. The spot rate this morning is U110.60/$, and early indications are the 90-day interest rates in the United States will rise
from their current level of 4.25% per annum. The U.S. Federal
Reserve Bank, the central bank, is worried about an unjustified
level of the U.S. stock market and has been publicly considering
raising interest rates 50 basis points (.5%). The 90-day forward
exchange rate quoted to Miyake by other Japanese banks are
all about the same, U109.80/$. The current 90-day yen interest rate is 0.5% per annum. Miyake has U400,000,000 at her
disposal.
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a. How much profit can Miyake make through covered interest
arbitrage?
b. What could Miyake do with her U400 million if she wished to
speculate through uncovered interest arbitrage? How much
profit could she expect to make?
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Problem 5. London and New York
Money and foreign exchange markets in London and New York
are very efficient. You have the following information:
Spot exchange rate
One-year treasury bill rate
Expected inflation rate
London
$1.6000/£
5.00%
2.00%
New York
£0.6250/$
6.00%
unknown
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Assume parity conditions hold:
a. Estimate inflation in the United States next year.
b. Estimate today’s one-year forward exchange rate between
pounds and dollars.
c. Diagram how the United Kingdom and the United States are
interrelated with respect to exchange rates, interest rates,
and inflation rates. Define on the diagram each of the parity
conditions which cause these theoretical linkages.
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Problem 6. Interest Arbitrage
Henri Jacque, an arbitrageur with Bank of Montreal, faces the
following Canadian dollar/U.S. dollar quotes:
Spot rate:
Six-month forward rate:
Six-month Canadian dollar interest rate
Six-month U.S. dollar interest rate:
C$1.4900/$
C$1.5100/$
7.50%
5.00%
Henri is authorized to use C$20,000,000 or its U.S. dollar equivalent. The ending profit, if any, should be realized in Canadian
dollars. How can he complete covered interest arbitrage? What
will be his profit?
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